《GRADUATE TEXTS IN MATHEMATICS 95:PROBABILITY》

Introduction1

CHAPTER ⅠElementary Probability Theory5

1.Probabilistic Model of an Experiment with a Finite Number of Outcomes5

2.Some Classical Models and Distributions17

3.Conditional Probability.Independence23

4.Random Variables and Their Properties32

5.The Bernoulli Scheme.Ⅰ.The Law of Large Numbers45

6.The Bernoulli Scheme.Ⅱ.Limit Theorems （Local，De Moivre-Laplace，Poisson）55

7.Estimating the Probability of Success in the Bernoulli Scheme68

8.Conditional Probabilities and Mathematical Expectations with Respect to Decompositions74

9.Random Walk.Ⅰ.Probabilities of Ruin and Mean Duration in Coin Tossing81

10.Random Walk.Ⅱ.Reflection Principle.Arcsine Law92

11.Martingales.Some Applications to the Random Walk101

12.Markov Chains.Ergodic Theorem.Strong Markov Property108

CHAPTER ⅡMathematical Foundations of Probability Theory129

1.Probabilistic Model for an Experiment with Infinitely Many Outcomes.Kolmogorov’s Axioms129

2.Algebras and σ-algebras.Measurable Spaces137

3.Methods of Introducing Probability Measures on Measurable Spaces149

4.Random Variables.Ⅰ.164

5.Random Elements174

6.Lebesgue Integral.Expectation178

7.Conditional Probabilities and Conditional Expectations with Respect to a σ-Algebra210

8.Random Variables.Ⅱ.232

9.Construction of a Process with Given Finite-Dimensional Distribution243

10.Various Kinds of Convergence of Sequences of Random Variables250

11.The Hilbert Space of Random Variables with Finite Second Moment260

12.Characteristic Functions272

13.Gaussian Systems295

CHAPTER ⅢConvergence of Probability Measures.Central Limit Theorem306

1.Weak Convergence of Probability Measures and Distributions306

2.Relative Compactness and Tightness of Families of Probability Distributions314

3.Proofs of Limit Theorems by the Method of Characteristic Functions318

4.Central Limit Theorem for Sums of Independent Random Variables326

5.Infinitely Divisible and Stable Distributions335

6.Rapidity of Convergence in the Central Limit Theorem342

7.Rapidity of Convergence in Poisson’s Theorem345

CHAPTER ⅣSequences and Sums of Independent Random Variables354

1.Zero-or-One Laws354

2.Convergence of Series359

3.Strong Law of Large Numbers363

4.Law of the Iterated Logarithm370

CHAPTER ⅤStationary （Strict Sense） Random Sequences and Ergodic Theory376

1.Stationary （Strict Sense） Random Sequences.Measure-Preserving Transformations376

2.Ergodicity and Mixing379

3.Ergodic Theorems381

CHAPTER ⅥStationary （Wide Sense） Random Sequences.L 2 Theory387

1.Spectral Representation of the Covariance Function387

2.Orthogonal Stochastic Measures and Stochastic Integrals395

3.Spectral Representation of Stationary （Wide Sense） Sequences401

4.Statistical Estimation of the Covariance Function and the Spectral Density412

5.Wold’s Expansion418

6.Extrapolation.Interpolation and Filtering425

7.The Kalman-Bucy Filter and Its Generalizations436

CHAPTER ⅦSequences of Random Variables that Form Martingales446

1.Definitions of Martingales and Related Concepts446

2.Preservation of the Martingale Property Under Time Change at a Random Time456

3.Fundamental Inequalities464

4.General Theorems on the Convergence of Submartingales and Martingales476

5.Sets of Convergence of Submartingales and Martingales483

6.Absolute Continuity and Singularity of Probability Distributions492

7.Asymptotics of the Probability of the Outcome of a Random Walk with Curvilinear Boundary504

8.Central Limit Theorem for Sums of Dependent Random Variables509

CHAPTER ⅧSequences of Random Variables that Form Markov Chains523

1.Definitions and Basic Properties523

2.Classification of the States of a Markov Chain in Terms of Arithmetic Properties of the Transition Probabilities p（n）ij528

3.Classification of the States of a Markov Chain in Terms of Asymptotic Properties of the Probabilities p（n）ii532

4.On the Existence of Limits and of Stationary Distributions541

5.Examples546

Historical and Bibliographical Notes555

References561

Index of Symbols565

Index569